Subgroups — #20297333

Order 7136.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813169	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₈₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813160	C₂	1	1	yes	maximal, characteristic, complemented
5	#16813158	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813153	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₄₄₆	—	2	1	no	—
9	Dic₄₄₆	C₂²	1	1	yes	characteristic
10	Dic₄₄₆	C₂²	1	1	yes	characteristic
11	C₂²⋊D₂₂₃	—	2	1	no	—
12	C₂×Dic₂₂₃	—	2	1	no	—
13	C₄×D₂₂₃	—	2	1	no	—
14	C₄×D₂₂₃	C₂²	1	1	yes	characteristic
15	C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	D₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₈×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₂₂₃	—	2	1	no	—
20	C₂×D₂₂₃	D₄	1	1	yes	characteristic
21	C₄×C₂₂₃	—	2	1	no	—
22	C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	Dic₂₂₃	—	2	1	no	—
24	Dic₂₂₃	—	2	1	no	—
25	Dic₂₂₃	D₄	1	1	yes	characteristic
26	C₂×C₂₂₃	—	4	1	no	—
27	C₂×C₂₂₃	C₂×D₄	1	1	yes	characteristic
28	D₂₂₃	—	2	1	no	—
29	C₂₂₃	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	223	1	no	maximal, Sylow
31	#41	—	223	1	no	—
32	C₂×Q₈	—	223	1	no	—
33	Q₁₆	—	223	1	no	—
34	Q₁₆	—	223	1	no	—
35	SD₁₆	—	223	1	no	—
36	SD₁₆	D₂₂₃	1	1	yes	characteristic, complemented
37	M₁₆	—	223	1	no	—
38	Q₈	—	446	1	no	—
39	Q₈	—	223	1	no	—
40	Q₈	—	223	1	no	—
41	Q₈	C₂×D₂₂₃	1	1	yes	characteristic
42	D₄	—	446	1	no	—
43	D₄	C₂×D₂₂₃	1	1	yes	characteristic
44	C₂×C₄	—	446	1	no	—
45	C₂×C₄	—	446	1	no	—
46	C₂×C₄	—	223	1	no	—
47	C₈	—	223	1	no	—
48	C₈	C₂×D₂₂₃	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	223	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	446	1	no	—
53	C₄	—	446	1	no	—
54	C₄	—	223	1	no	—
55	C₄	C₂²×D₂₂₃	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	446	1	no	—
58	C₂	D₄×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂³
Frattini	C₄	C₂²×D₂₂₃
Fitting	SD₁₆×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂×D₄
hypercenter	SD₁₆	D₂₂₃
hypocenter	C₂₂₃	#137
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. Q₈×D₂₂₃	C₂	semidirect
3. Dic₈₉₂	C₂	semidirect
4. #16813160	C₂	semidirect
5. #16813158	C₂	semidirect
6. #16813153	C₂	semidirect
7. SD₁₆×C₂₂₃	C₂	semidirect
29. C₂₂₃	#137	semidirect
36. SD₁₆	D₂₂₃	semidirect